Comparative analysis of the difference and probabilistic computational models to study partial differential equation of the elliptical type

Authors: Kryukov S.A.
Published in issue: #5(82)/2023
DOI: 10.18698/2541-8009-2023-5-894
Category: Mathematics | Chapter: Computational Mathematics
Keywords: temperature field simulation, elliptic equation, difference scheme, particle random walk, Dirichlet boundary conditions, probabilistic method, difference method
Published: 19.05.2023

The paper presents a comparative analysis of the difference and probabilistic computational models to study the heat transfer equation with the boundary conditions of the I kind. The first model was implemented by an explicit five-point scheme, and the second — by the random walk algorithm. The proposed approaches were confirmed by simulating the temperature field in a thin rectangular plate exposed to external radiation sources. During a series of experiments, it was found that the probabilistic method for determining the temperature value at a single point made it possible to obtain an approximate result from 2 to 200 times faster, depending on the number of irradiated particles and requird less memory compared to the difference method. It was established that, if the point under study was located close to one of the boundaries, the difference between results obtained during the algorithms operation was on average 1°C. And if its location was in the middle of the plate, the differences reached 3°C. The probabilistic model in most cases would lead to the incorrect result with a small number of the grid nodes. Meanwhile, the difference model demonstrates values that are more accurate and determines the thermal field completely.

References

[1] Egorov Yu.V., Shubin M.A. Linear partial differential equations. Foundations of the classical theory. Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya, 1988, vol. 30, pp. 5–255. (In Russ.).

[2] Savchenko A.O., Cavchenko O.Ya. Conducting body in an alternating magnetic field. Zhurnal tekhnicheskoy fiziki, 2015, vol. 85, iss. 7, pp. 8–12. (In Russ.).

[3] Samarskiy A.A., Nikolaev E.S. Metody resheniya setochnykh uravneniy [Methods for solving grid equations]. Moscow, Nauka Publ., 1978. (In Russ.).

[4] Ermakov S.M. Metod Monte-Karlo i smezhnye voprosy, ser. Teoriya veroyatnostey i ma-tematicheskaya statistika [Monte Carlo Method and Related Issues, ser. Theory of Probability and Mathematical Statistics]. Moscow, Nauka Publ., 1971. (In Russ.).

[5] Kuznetsov V.F. Reshenie zadach teploprovodnosti metodom Monte-Karlo [ Solution of heat conduction problems by the Monte Carlo method]. Moscow, IAE im. I.V. Kurchatova Publ., 1973, 21 p. (In Russ.).

[6] Saad Y. Iterative Methods for Sparse Linear Systems. SIAM, 2003, pp. 231–234. http://doi.org/10.1137/1.9780898718003